Rich parents are smarter, so their kids are smarter, so wealth is simply a confounder.
Smarter parents will on average have done better in their own education, thus on average leading to better paid jobs, leading to the ability to pay for private education.
Smarter parents will on average have smarter kids.
So I think some of reason 1 is valid.
You need to look at numbers applying from each sector and also numbers making the required grades, even if grades are adjusted to account for kids applying from less good schools.
You need to consider people self-selecting out from applying. There was a recent thread on here somewhere about 'will my poor DC feel out of place at Oxbridge'.
Both Oxford & Cambridge do a lot of outreach and want to take more kids from state schools. I agree they are not there yet, but progress is being made.